Assessment of load transfer characteristics
of a fiber-reinforced titanium-matrix composite
U. Ramamurty
*
Department of Metallurgy, Indian Institute of Science, C.V. Raman Avenue, Bangalore 560 012, India
Abstract
The extent of stress transfer that occurs around broken fibers dictates the longitudinal strength as well as the reliability of the
fiber-reinforced composites. The governing load transfer characteristics of a 32-vol% SiC fiber reinforced Ti-6Al-4V matrix composite (TMC) were investigated by recourse to uniaxial tension and 4-point bend flexure tests. Measured strengths and the variability in them are compared with the predictions of different load sharing models. The experimental tensile stress–strain response
indicates to localization in fiber failure and the local load sharing (LLS) model predicts the composite strength more closely than
a global load sharing (GLS) model. The flexure strengths are significantly larger than those predicted using the analytical models but
are similar to that obtained using simulations that incorporate stress gradients and LLS. Broader distribution in flexure strength is
also in agreement with simulation results, indicating that the LLS governs the strength of the TMCs. Implications of this observation are discussed. The TMCs are compared and contrasted with the Al-matrix composites in terms of their strength variability and
size-scaling.
Keywords: A: Metal matrix composites; Fibers; B: Strength; C: Stress transfer; Statistics
1. Introduction
The longitudinal strength and reliability of continuous fiber-reinforced metal matrix composites (MMCs)
are controlled by two factors: (i) the intrinsic strength
characteristics of the fibers, dictated by the flaw distribution, and (ii) the extent of stress re-distribution that occurs around fiber breaks and its effect on subsequent
fiber fracture [1]. The present article focuses on the latter
problem. When both the fiber-matrix interface and the
matrix itself are strong, the re-distribution of stress
may occur very locally around the break location, over
distances comparable to the fiber spacing. Conversely,
if either the interface or the matrix are sufficiently weak,
the re-distribution may occur equally over a longer
*
Tel.: +91 80 2293 3241; fax: +91 80 2360 0472.
E-mail address: ramu@met.iisc.ernet.in.
transverse length scale, spanning many a fiber-spacing.
This extreme situation is termed as global load sharing
(GLS). The other extreme, wherein the stress of a broken fiber is shed only to its immediate neighbors, is
one of strongly local load sharing (LLS). Intermediate
degrees of load sharing can be expected in most MMCs
of commercial interest.
The degree of load sharing plays a central role in the
sequence of fiber breakage during tensile straining and
hence on the composite strength and its variability.
Notably, under GLS conditions, the composite strength
is essentially deterministic, provided that the gauge
length is greater than the characteristic load transfer
length, dc, (defined in Section 2) and the number of fibers within the composite is sufficiently large (P103 to
104) [2]. In contrast, under strongly LLS conditions,
the fiber bundle strength is intrinsically stochastic and
dependent upon the volume of material under stress
[3]. In both cases, stochastic effects arise when the gauge
length is smaller than dc or when the number of fibers is
small.
The LLS behavior is exemplified by the Al2O3-fiber
reinforced Al alloys (without fiber coatings), described
in [4]. In these systems, the matrix bonds strongly to
the fibers. Moreover, for typical Al alloys (excepting
high purity Al itself), the yield strength of the matrix
is sufficiently high that the stress concentrations around
fiber breaks cannot be completely eliminated by the
attendant plastic straining in the vicinity of the break.
The localized nature of the load sharing manifests in
the volume-dependence of strength, as demonstrated
by tests performed in uniaxial tension as well as 3- and
4-point flexure. Despite the LLS conditions operative
in this system, the variability in the in situ fiber bundle
strength is remarkably small, with a Weibull modulus
of about 50 or, equivalently, a coefficient of variation
of about 2%.
The focus of the present paper is on a system comprising SiC fibers in a Ti alloy matrix (TMC). The SiC
fibers are coated with C in order to inhibit chemical
reactions between the fibers and the Ti matrix during
consolidation. The C coatings are weak and promote
interfacial debonding and sliding following fiber failure.
This process is expected to alleviate the stress concentrations associated with the fiber breaks and hence the
TMCs are considered, hitherto, as composites that obey
GLS with a deterministic strength [5]. Preliminary
experiments on a limited set of experiments, that too
on a relatively small specimen volume range, reinforce
this notion [6]. However, predictions made on composites that obey LLS suggest that for critical assessment
of the load transfer characteristics, experiments on specimens with very large differences in volumes (at least two
to three orders of magnitude difference) have to be conducted [7]. Furthermore, statistically significant number
of experiments for any given specimen volume is necessary to obtain information about the strength variability, which in turn can be used to assess the governing
load sharing mechanism.
The main objective of this work is to critically ascertain the load transfer characteristics between fiber and
the matrix in the TMC system and establish whether
the strength follows weakest link scaling laws, particularly in nonuniform loading conditions such as bending.
This is accomplished by comparing and analyzing the
strength and its variability in both uniaxial tension
and 4-point flexure. The experimental measurements
are analyzed in the context of existing models of fiber
bundle failure subject to various load transfer characteristics, ranging from GLS to strongly LLS. The notable
theoretical developments are attributable to Phoenix
and Raj [8], Curtin [1,2,7,9] and their co-workers. The
key features of the models are highlighted in the subsequent section. This is followed by a presentation of the
experimental measurements on the SiC/Ti composite
and the corresponding analysis.
2. Background
Relevant theories of fiber fragmentation and bundle
failure in unidirectionally reinforced MMCs are reviewed briefly, with emphasis on differences in behavior
under LLS and GLS conditions. Detailed review of this
topic is given by Curtin [1]. In both the cases, the extent
of load transfer along the fiber length is governed mainly
by the shear strength of the interface, s. In systems with
weak interfaces, s is controlled by the sliding resistance
of the interface; in systems with well bonded interfaces,
in contrast, it is controlled by the shear yield stress of the
matrix, sy. This shear resistance also plays an important
role in determining the extent of stress re-distribution to
neighboring fibers: GLS conditions are favored when s
is sufficiently low. For both GLS and LLS conditions,
the fiber bundle strength is dictated by a characteristic
transfer length, dc, given by [2]
!
1=m m=ðmþ1Þ
RS c
r0 RL0
dc ¼
¼
ð1Þ
s
s
and a characteristic strength, Sc,
1=ðmþ1Þ
r0 sL0
;
Sc ¼
R
ð2Þ
where R is the fiber radius, m is the Weibull modulus of
the fibers, and r0 is the reference strength corresponding
to a reference length, L0.
GLS assumes that the load lost by a fiber due to its
breakage and slippage is transferred equally to all the
unbroken fibers in the cross-sectional plane of the break.
For a composite whose length is much greater than 2dc
and contains very large number of fibers in the cross-section, Curtin [2] obtained analytical solutions for the in
situ bundle strength, rb, as
1=ðmþ1Þ 2
mþ1
rb ¼ S c
:
ð3Þ
mþ2
mþ2
Assuming an elastic, perfectly-plastic matrix, the postyield stress–strain response under isostrain condition,
applicable for stressing of the unidirectional composite
in the fiber direction, can be written as
mþ1 !
1 Ef e
ð4Þ
rðeÞ ¼ fEf e 1 þ ð1 f Þrym ;
2 Sc
where f is the fiber volume fraction, rym is the matrix
yield stress, and Ef and Em are the elastic modulus of
the fiber and the matrix, respectively. From Eq. (4),
the ultimate strength of the composite, ruc can be obtained as [1]
ruc ¼ fS c
2
mþ2
1=ðmþ1Þ mþ1
þ ð1 f Þrm
y :
mþ2
ð5Þ
Note that the ruc thus obtained is a deterministic quantity and does not scale with the size of the composite.
Pheonix and Raj [8] considered the transverse scale
of stress distribution to be much smaller than the
cross-sectional dimension of the composite (i.e., no
stress gradients within the vicinity of a broken fiber
were considered, instead all the fibers within the bundle
were assumed to carry the additional load shed on
them due to the broken fiber equally) and estimated
the composite strength and its variability using the
chain of bundles concept with weak-link scaling originally proposed by Gücer and Gurland [10]. Close
b , and Weibull
approximations of the mean strength, r
modulus, mb, of the in situ fiber bundle were obtained
as [8]
1
lnðln N Þ þ lnð4pÞ pffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffi
b ¼ C 1 þ
S c l þ c
r
2 ln N
mb
8 ln N
ð6Þ
and
pffiffiffiffiffiffiffiffiffiffiffiffi
l 2 ln N
mb ¼
þ 0:5½lnðln N Þ þ lnð4pÞ 4 ln N ;
c
ð7Þ
where C(Æ) is the gamma function, l* and c* are the mean
strength and standard deviation of the finite-sized fiber
bundle containing n fibers and N is the dimensionless
composite size. For a given composite of volume V, N
can be obtained by using the relation [1]
N¼
fV
:
0:4npR2 dc
ð8Þ
Predicting the strength of a composite that obeys
LLS by analytical means is a considerably more complicated problem, with the key task being the identification of the critical cluster of damaged fibers that
leads to instability in the composite component [1].
This requires coupling of statistics with mechanics
and close monitoring of the evolution of the fiber damage under the imposed LLS condition. Computer simulations of this problem by Zhou and Curtin [9],
discretized and by employing the lattice Greens function method with tunable load sharing, led to an analytical model that connects failure under LLS and GLS
and hence is capable of predicting the strength and its
variability through weak link scaling. Notably,
Ibnabdeljalil and Curtin [7] have shown that at some
critical cluster or ‘‘link’’ size in LLS, the failure probability under LLS is identical to that obtained under
GLS (i.e., Eqs. (6) and (7)) for the same size n. In
other words, the strength and reliability of composite
governed by LLS with a critical cluster size of nl (of
length equal to 0.4dc) will be exactly the same as that
of a composite governed by GLS with a finite-sized
fiber bundle of size nl. Once this identity is established,
all that remains to be done is the identification of nl
for a given LLS in a particular composite system
and then determine the size-dependent composite
strength and reliability using Eqs. (6)–(8). For very
strong LLS, Ibnabdeljalil and Curtin [7] have found
empirical correlations (on the basis of extensive numerical simulations) between nl and m as
nl ¼ 403m1:28 :
ð9Þ
In the following, the results obtained on TMCs are
analyzed within the framework of these models in order
to assess the load sharing characteristics in it.
3. Material and experiments
The material used in this study was a Ti-6Al-4V alloy
reinforced with 32 vol% of unidirectionally aligned
Sigma SiC fibers (fiber radius, R = 50 lm). The panel
consisted of 6 plies of fibers with a total thickness of
1.0 mm. A micrograph of the cross-section through the
composite is shown in Fig. 1. A particularly notable feature of the microstructure is the uniformity of the fiber
distribution. This leads to the very narrow distribution
in fiber volume fraction.
Uniaxial tensile tests were performed on straight
bars with a 6-mm width and 75 mm gauge length.
Stainless steel tabs were affixed to the specimen ends
using epoxy. Tensile strains were measured using strain
gauges that were bonded to the specimens. Four-point
flexure tests were performed on 6 mm wide specimens.
The loading direction was normal to the plane of the
composite panel. The inner and outer loading spans
were 9 and 27 mm, respectively. Strains were measured
on both the tensile and compressive faces using strain
gauges. In total, 9 tension and 20 flexure tests were
conducted.
The analysis of the strength measurements was
based in part on previous measurements of other
properties of the composite. These include the strength
distribution of the fibers (extracted from the composite) [11]; the sliding resistance of the interface, measured by fiber pushout as well as pullout [11,12];
and the residual thermal stresses measured by a matrix
Fig. 1. A cross-sectional view of the SiC fiber reinforced TMC.
Table 1
Relevant properties of fibers, matrix and composite
Property
Value
BP sigma fibers
Radius, R
Elastic modulus, Ef
Reference strength, r0
Reference length, L0
Weibull modulus, m
Mean strength of Fiber Bundle, l*
Standard deviation in fiber
bundle strength, c*
50 lm
380 GPa
1.47 GPa
1m
5.3
0.749Sc
0.0297Sc
Ti-6Al-4V alloy matrix
Elastic modulus, Ef
Yield stress,
Yield strain,
Composite
Number of fiber rows
Fiber volume fraction, f
Elastic modulus, Ec
Residual strain in the fibers
Sliding stress, s
Push-in measurement
Pull-out measurement
Characteristic strength, Sc
s = 130 Mpa
s = 60 MPa
Characteristic length, dc
s = 130 MPa
s = 60 MPa
114 GPa
820 MPa
0.72%
4. Results
Reference
[6]
[6]
[6]
[20]
6
0.326
200 GPa
0.152%
[13]
130 MPa
60 MPa
[5]
[11]
4.1. Stress–strain response
Typical tensile stress, r, and the corresponding
instantaneous tangent modulus, dr/de, variation with
tensile strain, e, are shown in Figs. 2(a) and (b), respectively. The composite starts to deform elastically upon
initial loading with an elastic modulus, Ec = 198 GPa,
a value consistent with that predicted using the ruleof-mixtures, Ec = fEf + (1 f)Em. The onset of yielding
occurs at 0.35% and the nonlinear behavior continues
until a strain of 0.55%. Upon further loading, the composite again deforms linearly but with a reduced slope
that is fEf, indicating that the fibers alone are carrying
all the additional load at this stage of deformation. Progressive fiber failure, as reflected by a slight decrease in
dr/de, initiates at about 1400 MPa and catastrophic
fracture ensues almost immediately (1482 MPa).
Typical nominal stress–displacement behavior under
flexure is shown in Fig. 3(a). Measured tensile and
4817 MPa
4261 MPa
1800
Experiment
1.8528 mm
3.5509 mm
1500
1200
900
600
300
(a)
0
250
Instantaneous Tangent Modulus,
dissolution technique [13]. The key properties are summarized in Table 1.
The average fiber volume fraction, f , was measured
in a number of fractured test specimens. The measurements were made by counting the total number of
fibers on the fracture surface, excluding the ones that
had been cut at the edges during machining, and subsequently calculating the area fraction from the fiber
diameter and the relevant cross-sectional area
(6 mm · 1 mm). Results of these measurements show
that the fiber volume fraction distribution is extremely
narrow, with the coefficient of variation of this distribution being 0.3%. This result is consistent with the
observed uniformity in the fiber distribution (Fig. 1).
The average fiber volume fraction, f is 0.326. Because
of the minimal variation in f, it imparting a statistical
distribution to the measured strength is assumed to
be negligible and hence is ignored in the subsequent
analysis.
Similarly, variability in the matrix strength affecting
the composite strength statistics is ignored in view of
the very low variability in the strength of ductile metals.
Also, the strength of the matrix is assumed to be independent of the volume of the material being tested. As
a result of these assumptions, the only parameter that
affects the size scaling and variability of the composite
strength will be the rb.
200
150
100
50
Experiment
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
(b)
Fig. 2. (a) Typical tensile stress, r, vs. strain, e, response of the TMC.
(b) Corresponding instantaneous tangent modulus, dr/de, curve
showing salient features of deformation. Predicted response using the
GLS model is also plotted for comparison.
2700
2400
2100
Load (N)
1800
1500
1200
900
600
300
0
0.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
Displacement (mm)
(a)
2500
Nominal Stress, σ (MPa)
Compressive Face
Tensile Face
2000
1500
2450
2400
2350
1000
2300
2250
2200
500
2150
2100
0
0.0
(b)
nal stress calculated on the basis of elastic beam theory
overestimates the true stress acting along the tensile face.
Another reason for the high strength in flexure could be
the high stress gradients effecting the fiber
fragmentation.
Note from Fig. 3(b) that at high stresses (close to the
first peak stress) considerable noise in both the tensile
and compression strain data is seen. Closer examination
(inset of Fig. 3(b)) reveals that it is due to discrete strain
jumps of about 0.16%. (Aside, distinct ‘‘clicking’’
sounds at higher stresses could be heard while conducting these tests.) Clearly, these strain jumps are a result of
the fiber fragmentation. The following are some of the
observations.
2050
0.5
1.0
1.5
2.0
Strain, ε (%)
Fig. 3. (a) Typical load–displacement behavior of the TMC under
flexure. (b) Tensile and compressive strains plotted against the nominal
stress. Inset shows a magnified view of the fiber failure regime.
compressive strains are plotted against nominal stress
(calculated by using Euler–Bernoulli beam theory for
elastic materials) in Fig. 3(b). A couple of differences between the tensile and flexure responses are noteworthy.
The load carrying capacity of the composite remains significant even after the first instability point. This discontinuous yet prolonged load carrying capacity beyond the
first peak is due to initiation, propagation, and arrest of
cracks caused by the bridging action of the fibers [14].
The first peak in the nominal stress, which also happens
to be the maximum stress carried by the specimen, is taken as the strength of the composite.
The second observation pertains to the strength and
failure strain (strain corresponding to the first peak) of
the composite under flexure. They are considerably
higher (by about 65%) when compared to their respective values in tension. An extrinsic cause for this anomaly is the macroscopic stress redistribution due to the
plastic flow of the matrix starting at the tensile face.
Due to this nonlinear stress–strain response, the nomi-
(1) The first strain jump occurs at a nominal stress of
2084 MPa, followed by considerable fiber failure
activity between 2300 and 2350 MPa.
(2) Above the stress level of 2370 MPa, the strain
bursts completely stop and further stress build up
without any softening is seen. This indicates that
the fiber fragmentation in the first row has saturated and the second row of fibers starts experiencing the higher stresses. Furthermore, residual fiber
pull out of the fragmented fibers aids in additional
load carrying capability. This observation clearly
implies that the saturation fragmentation of the
first row of fibers in itself is insufficient to cause
the specimen to fracture.
(3) Relatively muted strain bursts are seen again
above the stress level of 2400 MPa, indicating that
the second row of fibers have started fragmenting.
Since the fragmenting fibers are farther away from
the strain gauge on the tensile face, the amplitude
of the strain bursts is relatively smaller.
4.2. Strength variation
Fig. 4 shows the variation in tensile and flexure
strength of the composite, plotted against the cumulative probability of failure, P, calculated using the
relation
P ¼ ði 0:5Þ=I
ð10Þ
where i is the ascending rank of a particular data point
in the data set of size I. The tensile strength distribution
is very narrow, with a range of only 40 MPa or 2.6%.
c , is 1482 MPa
The mean strength of the composite, r
with a standard deviation of 14.2.
The
uncertainty in
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
the standard deviation ð1= 2ð9 1Þ ¼ 0:25Þ is
3.6 MPa. The mean tensile failure strain is 1.00% with
a standard deviation of 0.0405%.
The strength distribution in flexure is much broader
(total range is 400 MPa or 16%) than that in tension.
c ¼ 2523 MPa with a standard deviation of
Here, r
m V r c
;
P ðrÞ ¼ 1 exp V 0 r0
Cumulative Failure Probability, P
1.0
Tension
Flexure
0.8
ð11Þ
where V is the specimen volume, V0 is the reference volume, r0 is the reference strength and mb is the Weibull
modulus of the fiber bundle. Extracted values of mb
are listed in Tables 2 and 3.
0.6
0.4
5. Discussion
0.2
5.1. Uniaxial tension
0.0
1200
The axial tensile stress–strain response, rc(e), of a unidirectional TMC can be modeled using the rule-of-mixtures [5,15]
1400
1600
1800
2000
2200
2400
2600
2800
Composite Strength, σc (MPa)
rc ðeÞ ¼ f rf ðeÞ þ ð1 f Þrm ðeÞ;
Fig. 4. Variation in tensile and flexure strengths of the composite
plotted against the cumulative probability of failure, P.
97.4 ± 15.8 MPa. Higher strength and broader distribution of it vis-à-vis tensile test results confirm to the expected trends of higher strength and larger scatter in
quasi-brittle specimens at smaller volumes. Average value of the strain to failure on the tensile face of the flexure specimen is 1.6921% with a standard deviation of
0.0646%.
Distribution in rb, obtained by subtracting the matrix
contribution from ruc , are fitted with Weibull distribution function,
ð12Þ
where rf(e) and rm(e) represent the average axial stresses
in the fiber and the matrix, respectively. The matrix is
assumed to be elastic-perfectly plastic, reasonable for
Ti-alloys. Hence,
rm ðeÞ ¼ Em e
e 6 eyc ;
ð13aÞ
¼ Em eyc
e P eyc
ð13bÞ
eyc
with being the composite yield strain. Note that in the
absence of any residual strains, eyc eym , the unconstrained matrix yield strain. However, residual strains
arise upon cooling the composite from the processing
temperature because of the large thermal expansion mis-
Table 2
Experimental and predicted tensile strength and its Weibull modulus of the TMC
Number of Links
Mean composite strength
(MPa)
Fiber bundle Weibull
modulus
s (MPa)!
130
60
130
60
130
60
GLS (n = 1)
GLS (finite n)
LLS
LLS (periodic BC)
Experiment
–
101
525
–
–
53
273
–
1656
1609
1585
1572
1529
1496
1474
1494
1
69
79
129
1
65
75
107
–
1482 ± 14
68 ± 10
23 ± 4 (failure strain)
Table 3
Experimental and predicted in situ bundle strengths and its Weibull modulus for the 4-point bend tests
Number of links
Mean in situ fiber bundle
strength (MPa)
Fiber bundle Weibull
modulus
s (MPa)!
130
60
130
60
130
60
GLS (n = 1)
GLS (finite n)
LLS
LLS (simulation)
Experiment
–
12
10
–
–
6
5
–
3385
3301
3362
5070
2994
3018
3031
4484
1
52
51
35
1
46
43
32
–
4163
16 ± 1
54 ± 9 (failure strain)
match between the fibers and the matrix [13]. In the asprocessed state, the matrix is under tension whereas the
fibers are in compression. Because of the preexisting axial tensile residual strains, eRm , in the matrix, its tensile
yield strain decreases by that amount. Thus,
eyc ¼ eym eRm :
ð14Þ
Residual strain in the fibers, measured in this TMC
using the selective etching technique is 0.15% [13].
Corresponding residual strains in the matrix (calculated)
are 0.27%. From Fig. 2(b), it is seen that eyc 0:45%.
Therefore, eym 0:72 which is similar to the typical values
for the yield strain of Ti-4Al-6V alloy of 0.7%. Note
also that the matrix yield strength, rym (estimated by
using an Em of 114 GPa) 820 MPa, is also consistent
with those typically reported for Ti-6Al-4V alloys [15].
The predicted post-yield stress–strain response of the
composite, using Eqs. (3) and (12), is plotted in Fig. 2.
Here, the fiber failure statistics measured by Weber et
al. [5] on fibers extracted from this TMC by dissolving
the matrix away and the fiber sliding resistance measured using fiber push-out tests (s = 130 MPa) are used.
However, push-out tests tend to overestimate the value s
because of the Poisson expansion of the fibers during the
test. Pull-out tests do not suffer from this drawback and
hence give a better estimate of the value for s since they
represent the stress state in the fiber during a tensile test.
Pull-out tests on a slightly different TMC by Walls and
Zok [11] yield a s of 60 MPa. Predictions made using
this s value are also tabulated.
As seen from Fig. 2(b), the model predictions are
commensurate with the experimental trends. However,
a couple of differences are noteworthy.
• Although the predicted and experimental strengths
appear to be in good agreement, the failure strains
are not in such a good agreement (10% difference).
One of the reasons for the difference in failure strains
could be a faster localization (or clustering) of fiber
breaks. From Fig. 2(b), a small drop in dr/de just
before fracture is noticeable. Thus far the experimental trends are commensurate with the GLS model predictions; indicating a small amount of non-interactive
fiber failure. However, catastrophic fracture ensues
much before the experimental dr/de reduces to 0,
indicating that localization initiates readily in this
composite.
• The model predicts a sharp drop in dr/de at the composite yield point whereas the experimental results
show a gradual transition. This discrepancy arises
because the model assumes the matrix to be elasticperfectly plastic and that the residual stresses are distributed uniformly. However, in reality, the Ti alloys
exhibit a slight hardening behavior [16]. Variation in
the residual stresses as a function of distance away
from the fiber could also lead to a gradual yielding
behavior [17]. Since our primary interest is in the near
fracture regime, which occurs at a much higher stress
level beyond yielding of the composite, we ignore the
gradual transition at yield.
Turning attention to the tensile strength and its variability, experimentally measured mean composite
strength and fiber bundle Weibull modulus are compared with those predicted using different models in
Table 2. For predicting the strength using the GLS
model of Phoneix and Raj [8], a priori knowledge of
the characteristic fiber bundle size, n, is necessary.
Here, we consider n to be equal to the number of fibers
in the tensile specimen cross-section (=240). Note that,
this approach reduces the Phoneix and Raj model, in
effect, to a GLS model corrected for finite bundle sizes.
All the predictions are made using two values of s, 130
and 60 MPa, obtained using push-out and pull-out
techniques.
It is seen from Table 2 that all the predicted mean
strengths using s = 130 MPa are higher than the experimentally measured value. For the case of s = 60 MPa,
Phoneix and Raj model as well as the LLS model predict
a mean strength that is very close to the experimentally
measured strengths. All the predicted mb are very high,
65 and above, suggesting a narrow distribution in the
strength, consistent with the experimental observation.
In particular, both the LLS model and the GLS model
with finite specimen size correction predict an mb that
is close to the experimentally measured value of 68. It
is interesting to note that the Weibull modulus for the
composite failure strain in tension is much smaller,
23 ± 4. Because of the gradual ‘‘flattening’’ of the
stress–strain curve predicted by the GLS model at its
maximum (see Fig. 2(a)), the distribution in failure stresses will be narrower than that in failure strains. Clearly,
a larger distribution in the latter implies to localization
(as exemplified by the experimental dr/de data in Fig.
2(b)) and hence it is reasonable to conclude that LLS
model is more appropriate for describing the strength
statistics of the TMC.
5.2. Flexure response
The flexure strengths obtained are analyzed to extract
rb, with the following two approaches.
1. Stress partitioning: Linearly partition the applied
bending moment, M, into two components, Mf and
Mm, bending moments pertaining to the fibers and
the matrix respectively. Assuming that the elastic-perfectly plastic matrix is fully yielded (reasonable, considering the fact that the failure strain of the flexure
specimens is 2.5 times the matrix yield strain),
Mm = 1.5ME, where ME is the maximum elastic
moment that the matrix can support and taking the
yield stress of the matrix as 820 MPa, the mean in situ
fiber bundle strength (extracted from the average flexure strength of 2520 MPa) is obtained as 5184 MPa.
2. Strain partitioning: Alternatively, assume a linear
strain distribution through the composite thickness
and hence calculate the strain in the first row of the
fibers to be equal to 0.78 times the strain on the tensile face, yielding an in situ bundle strength of
4980 MPa.
and length equal to 2dc with periodic, semi-free and free
boundaries, that the mean fiber bundle strengths are
well within a standard deviation of one another and
hence concludes that for a given m, the effect of boundary conditions on the fiber bundle strength is statistically
insignificant. This is possibly the reason for the observed
similarity in the simulation and analytical predictions.
Turning attention to the flexure tests, it is seen from
Table 3 that the analytical and simulated LLS results
For predicting rb using the various models presented
in Section 2, we recognize that in essence the 4-point
bend tests are tantamount to tensile testing of a composite of length 9 mm (inner span of the flexure specimen)
and 6 mm (width of the specimen) in tension with single
row of fibers in it. Predicted rb and its variability (in
terms of the Weibull modulus) are listed in Table 3.
As seen, both the values of the bundle strengths extracted from the experimental flexure results through
the simple analyses are very large when compared to
those predicted. This is clearly a consequence of the
inability to analytically model the stress distribution
due to matrix yielding as well as fiber breakage in flexure
loading. Exact analysis of the flexure test results and
comparison of the experimental and predicted stress–
strain behavior is complicated by the following factors.
1. Relatively large size of the fibers vis-à-vis the specimen thickness, which leads to a large strain gradient
across the fibers. Also, the small number of fibers
across the specimen thickness (6) renders the application of continuum elasto-plastic stress analysis of the
flexure beams difficult.
2. High strength of the matrix and hence significant contribution of it to the overall strength of the composite. Incorporating plasticity of the matrix at the
same time as stochastic fiber bundle response to
extract the stress state at failure analytically is
difficult.
Foster [18] has performed numerical simulations on
identical geometries to the tensile and 4-point bend flexure tests under LLS conditions, incorporating periodic
boundary conditions (mimicking the tensile samples
with six fibers in the cross-section) as well as stress gradients (mimicking the flexure specimens). The failure criterion for bending was considered to be formation of a
half-plane of broken fibers below the center of the composite. Fosters predictions, based on 50 simulations of
tensile geometry and 35 simulations of flexure geometry,
are also given in Tables 2 and 3, respectively. It is seen
that from Table 2 that the results of the simulations under LLS and with periodic boundary conditions are similar to those predicted using the analytical LLS model.
Foster observed, on the basis of hundreds of simulations
performed on composites with 4 · 26 lattice of fibers
Fig. 5. (a) Plot of cumulative fiber failures vs. applied fiber bundle
stress (normalized with the characteristic bundle stress, Sc) under 4point bending in a LLS simulation. (b) Corresponding evolution of
composite failure showing fiber breaks in top three layers of the
composite that is subjected to increasing 4-point bending stress.
(the latter incorporating the stress gradients associated
with the flexure) diverge significantly. Furthermore,
the simulation results are in very good agreement with
the experimental results, especially the predicted mc.
Note that the LLS model with a s = 60 MPa also appears to predict a mc that is reasonably close to the
experimentally observed mc, the latter value unaffected
by the complications in the load redistribution.
A key observation made during the LLS simulation of
the flexure test by Foster [18] is that the composite is
capable of sustaining much more damage even after the
breakage of first layer of fibers. Fig. 5(a) shows the
cumulative fiber failures plotted against the fiber bundle
stress (normalized with the characteristic stress, Sc) in a
typical simulation of the flexure test. Each datum on this
plot represents a fiber break. Fig. 5(b) shows the corresponding evolution of the damage in the top three layers
of the flexure specimens. (Recall that the specimen has 6
fiber rows and hence Fig. 5(b) represents the fiber rows
on the tensile side of the specimen.) As seen from Fig.
5(a), failure of the first row of fibers in the composite
specimen occurs at 0.89Sc with approximately 200 fiber
breaks. In general these simulations are consistent with
the experimental observations of Fig. 2(b). While this
stress itself is much higher than those predicted using
the GLS and LLS models (0.70Sc), further ‘‘hardening’’ of the composite is seen and it appears that the
stress-fiber break plot flattens out asymptotically until
the fiber breaks have reached the midsection of the specimen (Fig. 5(b)). While the exact failure criterion beyond
this point is difficult to implement, it is clear from Fig.
5(a) that the high stress gradients present in the flexure
specimens leads to considerably higher strength, broadly
corroborating with the experimental results. These simulation results also imply that the LLS model is far better
equipped, albeit requiring simulations of the exact geometry at times, to capture the actual material behavior.
5.3. TMC vs. AMC
The Ti-matrix composite examined in this work is
distinctly different from the 70 vol% Al2O3 fiber reinforced Al-matrix composite (AMC) that we have examined in an earlier paper. In particular, the TMC differs
from the AMC in three important respects as following.
• First, the yield strength of the Ti-alloy represents a
significant fraction (0.5) of the ultimate tensile
strength (UTS) of the composite. Consequently, the
tensile response of the composite is bilinear, with
the change in slope occurring at roughly one half of
the failure strain. An additional consequence is the
development of non-linearity in the stress distribution
across a flexure specimen following matrix yielding.
Such effects must be taken into consideration in evaluating the true tensile stress within the fibers at fail-
ure. In contrast, in the AMC, the yield strength of
the matrix alloy is negligible (<5% of the UTS). Consequently, the tensile response is essentially linear up
to fracture, and, in flexure, the stress distribution
remains linear [4].
• Second, the diameter of the SiC fibers is about an
order of magnitude larger than that of the Al2O3
fibers. The fiber size has two implications. (i) Under
flexural loading of the TMC, substantial strain gradients occur over distances comparable to the fiber
spacing. Moreover, the fibers closest to the tensile
face experience a strain that is considerably lower
than that in the matrix on the tensile face. In the
AMC, the gradients that occur about a fiber spacing
are negligibly small. (ii) The number of fibers found
in a typical TMC tensile specimen is only 200, compared to 20,000 for a similarly-sized AMC specimen
(the number ratio scaling with the inverse square of
the ratio of fiber diameters). In the TMC flexural
specimens, the effective number of fibers controlling
strength is lower yet (<100). Such small numbers
can cause additional stochastic effects in the fiber
bundle strength.
• Third, fiber coatings similar to that of C coating of
SiC fiber in the TMC have not been utilized extensively in the AMC because of their role in reducing
the transverse strength to extremely low levels. In
the absence of coatings, the AMC interface is strong,
leading to strongly LLS characteristics.
Despite these significant differences, TMCs and
AMCs both obey the LLS model predictions with the
size-scaling of their strength and reliability conforming
to the analytical models developed by Ibnabdeljalil
and Curtin [7]. This is perhaps due to the relatively high
value of s in both the cases, given by the interfacial sliding resistance for the TMC and by the matrix shear yield
stress for the AMC. Another factor could be the large
size of the fibers vis-à-vis the specimen size, leading a
stress concentration analogous to that present at a notch
as per fracture mechanics, leading to failure localization.
However, the spatial as well as the specimen-to-specimen variations is fiber volume fraction are the additional considerations that have to be taken into
consideration for understanding the AMC strength variability [4,19].
5.4. Implications
Experimental results presented in this paper clearly
demonstrate that there is an intrinsic size-scaling to the
strength of the fiber reinforced MMCs. This size-scaling
that arises due to the LLS conditions that prevail in these
composites is particularly important when a large MMC
components strength has to be predicted on the basis of
strength data generated using small coupons such as
those used in our flexure tests on TMCs. Comparisons of
the experimental data with predictions made using available models clearly show that the GLS models, while successful on occasion, can lead to erroneous predictions.
For example, the GLS model predicts a mean strength
of 1529 MPa for tensile samples used in this study and
hence appears to be in reasonable accordance with the
experimentally measured value of 1482 MPa (only 3%
difference). The observed variability in tensile strength
also appears to show that it is a deterministic quantity.
However, for a component containing 100,000 links (a
connecting rod with a 25-mm diameter and 20 cm
length), the TMC will have 1416 MPa strength (as per
LLC model), and hence the GLS would be over-predicting the strength by 10%. For very small components
sizes, GLS not only underestimates the strength but also
predicts a higher reliability, which can lead to non-conservative designs. The GLS model also suffers from the
fact that it overestimates that failure strains, which have
important implications in fracture related properties of
the composite such as its notch sensitivity.
The Phoneix and Raj model, while appears to be as
good as the LLS model in terms of predicting both the
tensile and flexure strengths as well as their variability,
suffers from the implicit drawback that it does not offer
any means to estimate the characteristic bundle size. The
approach we have used in this work to employ this model, namely taking n to be equal to the number of fibers in
the specimen cross-section, will not work when the specimen size is large. As n ! 1 the predictions will be identical to that made using the GLS model.
In view of the above constraints suffered by the GLS
model and its variants, the LLS model appears to best
model to describe the mechanical behavior of the
TMC investigated in this work as well as the AMCs
examined elsewhere [4]. Furthermore, good agreement
between the experimental results and LLS model predictions imply that the load sharing between the fragmented fibers and its neighbors tends to be local in
nature. However, it should be noted that these analytical
models could only capture the strength of the composites when there is homogeneous loading (i.e., where
there is uniform stress such as the uniaxial tension) with
a reasonable degree of accuracy. Among the models
available, the analytical LLS model appears to give most
accurate results. However, it appears that for complex
geometries and large stress gradients (such as flexure
loading), computational simulation is the only recourse
that is available. This is particularly true when the complexity gets compounded by the discreteness in the material system (like the presence of only a few fiber rows
across the specimen thickness). This is precisely the reason for the lack of complete failure in Fosters simulations, only a stress region over which damage
accumulates very rapidly, after which it slows down
due to the constraints of equilibrium in the discrete sys-
tem. What exactly is the failure criterion that has to be
employed in such scenarios for predicting the fracture
of the structure thus remains an open question.
In closing, it should be noted that the LLS model employed in this paper for the analysis of the experimental
results is that developed Curtin and coworkers who treat
the statistics of failure in terms of a chain of Gaussian
bundles. Phoenix and Beyerlein [21] treat this problem
differently, which leads to a fundamentally different
form of distribution altogether, at least for moderate
to larger fiber Weibull modulus values. However, while
mathematically different, the predictions of these two
LLS models are very similar.
6. Conclusions
The strength and its variability of a 32-vol% SiC fiber reinforced Ti-6Al-4 V matrix composite, measured
by conducting both uniaxial tension as well as 4-point
bend flexure tests, suggest that the final failure of this
composite is governed by the local load sharing. The
experimental tensile stress–strain response indicates to
localization in fiber failure as expected under LLS conditions. The flexure strengths are significantly larger than
those predicted using the analytical models but are similar to that obtained using simulations that incorporate
stress gradients and LLS. Similarly, distribution of the
flexure strengths is broader, in agreement with simulation results. This work also highlights the possible problems involved in deciphering the data generated by
flexure testing of composites and in turn connecting them
with those tests conducted in a much simpler stress states.
Acknowledgement
It is indeed a pleasure to acknowledge Frank Zok of
UCSB and Bill Curtin of Brown University for many
useful discussions and insights. Thanks are also due to
Glenn Foster, formerly at VPISU and now at UIUC,
for his permission to use Figs. 5(a) and (b) (of this paper) from his MS thesis.
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